Find Critical Value Za/2 with Statcrunch

Professor Jen
7 Oct 201803:53
EducationalLearning
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TLDRThe video script demonstrates how to find the critical Z-value for a 91% confidence level using StatCrunch software. It explains that the Z-values correspond to the standard normal distribution with a mean of 0 and standard deviation of 1. The process involves using the normal distribution calculator in StatCrunch, setting the confidence level to 91%, and computing to find the critical values of -1.6954 and 1.6954. Since the distribution is symmetric, only the positive value, 1.6954, is needed for Z-alpha over 2. The video concludes with rounding this value to two decimal places, resulting in 1.70 as the final critical Z-value.

Takeaways
  • 🔍 The video discusses finding the critical value Z alpha over 2 for a specific confidence level using technology.
  • 💻 The presenter uses StatCrunch software to demonstrate the process of finding the critical value.
  • 📈 The Z values are associated with the standard normal distribution, which is a bell-shaped curve with a mean of 0 and a standard deviation of 1.
  • 📊 The presenter opens the normal distribution calculator within StatCrunch to find the critical value.
  • 🔄 The confidence interval is represented as a range with a lower and upper limit, which separates the interval in the middle.
  • 🔢 A confidence level of 91% is used in the example, indicating the area between the lower and upper limits of the interval.
  • 📝 The area corresponding to the confidence level is entered as a decimal (0.91) into the calculator.
  • 📉 After computation, the calculator provides the critical values that separate the bottom and top of the shaded area under the curve.
  • 📌 The critical values given are -1.6954 and +1.6954, representing the symmetric nature of the standard normal distribution.
  • 📝 The question asks for Z alpha over 2, which refers to the positive critical value only, due to the symmetry around the mean.
  • 🔧 The final step involves rounding the positive critical value to two decimal places, resulting in 1.70 as the rounded critical value.
Q & A

  • What is the main topic of the video script?

    -The main topic of the video script is finding the critical value Z alpha over 2 for a specific confidence level using StatCrunch software.

  • What is the confidence level discussed in the script?

    -The confidence level discussed in the script is 91%.

  • What does Z alpha over 2 represent in the context of the script?

    -Z alpha over 2 represents the critical value from the standard normal distribution that corresponds to a particular confidence level.

  • What is the standard normal distribution?

    -The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1, characterized by its symmetric bell-shaped curve.

  • How does the script suggest finding the critical value using technology?

    -The script suggests using StatCrunch software to find the critical value by navigating to the normal distribution calculator within the program.

  • What is the significance of the confidence interval in the script's context?

    -The confidence interval provides a range of values that will contain the true population parameter with a certain level of confidence, in this case, 91%.

  • What does the script mean by 'between' in the context of the normal distribution calculator?

    -In the context of the normal distribution calculator, 'between' refers to the option that calculates the area under the curve between two specified values, which corresponds to the confidence interval.

  • How does the script describe the process of entering the confidence level into StatCrunch?

    -The script describes entering the confidence level as a decimal (0.91) into the designated area box in the normal distribution calculator in StatCrunch.

  • What are the critical values obtained for the 91% confidence level in the script?

    -The critical values obtained for the 91% confidence level are -1.6954 and +1.6954.

  • Why does the script only provide the positive critical value in the final answer?

    -The script only provides the positive critical value because the standard normal distribution is symmetric, and the question specifically asks for Z alpha over 2, which corresponds to the right-hand side of the curve.

  • What rounding rule is applied to the final critical value in the script?

    -The rounding rule applied is to round up when the third decimal place is 5 or more, resulting in the final critical value of 1.70.

Outlines
00:00
📊 Finding the Critical Z Value for a Confidence Level

The speaker discusses the process of determining the critical Z value, Z alpha over 2, associated with a specific confidence level using technology. They choose to use the StatCrunch program and navigate to the normal distribution calculator within it. The objective is to find the Z value that corresponds to a 91% confidence level. The speaker explains that the standard normal distribution, which is symmetric and bell-shaped with a mean of 0 and a standard deviation of 1, is used for this calculation. They input the desired confidence level into the calculator and obtain the critical values, which are -1.6954 and 1.6954. Since the question asks for Z alpha over 2, only the positive value is needed. The speaker rounds the positive value to two decimal places, resulting in 1.70 as the final critical value.

Mindmap
Keywords
💡Critical Value
A critical value is a threshold in statistical analysis that is used to determine the significance of results. In the context of the video, the critical value Z alpha over 2 is the value from the standard normal distribution that corresponds to the tail end of the distribution and is used to calculate confidence intervals. The script explains that for a 91% confidence level, the critical value is approximately 1.6954, which is the point at which the remaining 9% of the data is found in the tails of the distribution.
💡Confidence Level
The confidence level is a percentage that represents the level of certainty one has in the results of a statistical analysis. In the video, a 91% confidence level is mentioned, which means that if the same statistical procedure is repeated multiple times, the true population parameter would fall within the calculated interval 91% of the time. The script demonstrates how to find the critical value associated with this confidence level using the normal distribution.
💡Z Score
A Z score is a standard score that indicates how many standard deviations an element is from the mean. In the video, Z scores are used to represent data points in the standard normal distribution, which has a mean of 0 and a standard deviation of 1. The critical value Z alpha over 2 is a specific Z score that helps in determining the confidence interval.
💡Normal Distribution
A normal distribution, also known as a Gaussian distribution, is a probability distribution that is characterized by its symmetric bell-shaped curve. The video discusses the standard normal distribution, which is a normal distribution with a mean of 0 and a standard deviation of 1. The critical values are found within this distribution to establish the confidence interval.
💡Standard Normal Distribution
The standard normal distribution is a specific type of normal distribution with a mean of 0 and a standard deviation of 1. It serves as a basis for comparing other normal distributions. The video uses the standard normal distribution to find the critical Z value for a given confidence level.
💡StatCrunch
StatCrunch is a statistical software program used for data analysis. In the video, the presenter uses StatCrunch to find the critical Z value for a 91% confidence level. It is demonstrated how to navigate the program and use its normal distribution calculator to find the necessary value.
💡Area Under the Curve
In the context of a normal distribution, the area under the curve represents the probability of a data point falling within a certain range. The video explains that a 91% confidence level corresponds to an area of 91% under the curve, with the remaining 9% in the tails of the distribution.
💡Confidence Interval
A confidence interval is a range of values, derived from a statistical model, that is likely to contain the value of an unknown parameter. It is expressed as an interval between a lower and upper bound. The video demonstrates how to calculate the bounds of a 91% confidence interval using the critical Z value.
💡Symmetric Curve
A symmetric curve is one that is mirror-imaged across its vertical axis. In the video, the standard normal distribution is described as having a symmetric bell-shaped curve, which means that the positive and negative tails are mirror images of each other, with the critical values being the same magnitude but opposite in sign.
💡Decimal Places
Decimal places refer to the digits that follow the decimal point in a number. In the video, when rounding the critical value to two decimal places, the presenter notes that the third decimal place (a 5) requires rounding up, resulting in the final critical value of 1.70.
💡Rounding
Rounding is the process of adjusting a number to the nearest value with a certain number of decimal places. The video demonstrates rounding the critical value from 1.6954 to 1.70 by considering the value in the third decimal place, which dictates the rounding rule.
Highlights

Introduction to finding the critical value Z alpha over 2 using technology.

Using StatCrunch to calculate the critical value for a specific confidence level.

Understanding the concept of Z values and their relation to the standard normal distribution.

Explanation of the standard normal distribution's properties: mean of 0 and standard deviation of 1.

Navigating the StatCrunch interface to access the normal distribution calculator.

Selecting the 'between' option in the normal distribution calculator for confidence intervals.

Setting the confidence level to 91% in the calculator to match the problem's requirements.

Entering the confidence level as a decimal in the calculator.

Computing the critical values using the calculator and interpreting the results.

Identifying the critical values as -1.6954 and 1.6954 from the standard normal distribution.

Explanation of the symmetric nature of the standard normal distribution and its implications for critical values.

Focusing on the positive critical value for the Z alpha over 2 calculation.

Rounding the critical value to two decimal places as per the question's instructions.

Final determination of the critical value as 1.70 after rounding.

Summary of the process for finding the critical value Z alpha over 2 for a 91% confidence level.

Transcripts

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Confidence IntervalsCritical ValueZ-ScoreStatCrunchNormal DistributionStatistical AnalysisData ScienceEducational ContentTutorial GuideMathematicsResearch Method